Thus it is clear that the movers are substances, and that one of them is first and another second and so on in the same order as the spatial motions of the heavenly bodies.As regards the number of these motions, we have now reached a question which must be investigated by the aid of that branch of mathematical science which is most akin to philosophy, i.e. astronomy; for this has as its object a substance which is sensible but eternal, whereas the other mathematical sciences, e.g. arithmetic and geometry, do not deal with any substance. That there are more spatial motions than there are bodies which move in space is obvious to those who have even a moderate grasp of the subject, since each of the non-fixed stars has more than one spatial motion.As to how many these spatial motions actually are we shall now, to give some idea of the subject, quote what some of the mathematicians say, in order that there may be some definite number for the mind to grasp; but for the rest we must partly investigate for ourselves and partly learn from other investigators, and if those who apply themselves to these matters come to some conclusion which clashes with what we have just stated, we must appreciate both views, but follow the more accurate.

Eudoxus^{1} held
that the motion of the sun and moon involves in either case three
spheres,^{2} of which
the outermost is that of the fixed stars,^{3} the second revolves in the circle
which bisects the zodiac,^{4}
[20]
and the third revolves in a
circle which is inclined across the breadth of the zodiac^{5}; but the circle in which the moon moves is inclined at
a greater angle than that in which the sun moves.And he held that the motion of the
planets involved in each case four spheres; and that of these the
first and second are the same^{6} as before (for
the sphere of the fixed stars is that which carries round all the
other spheres, and the sphere next in order, which has its motion in
the circle which bisects the zodiac, is common to all the planets);
the third sphere of all the planets has its poles in the circle which
bisects the zodiac; and the fourth sphere moves in the circle inclined
to the equator of the third. In the case of the third sphere, while
the other planets have their own peculiar poles, those of Venus and
Mercury are the same.

Callippus^{7} assumed the same arrangement of the
spheres as did Eudoxus (that is, with respect to the order of their
intervals), but as regards their number, whereas he assigned to
Jupiter and Saturn the
same number of spheres as Eudoxus, he considered that two further
spheres should be added both for the sun and for the moon, if the
phenomena are to be accounted for, and one for each of the other
planets.

But if all the spheres in combination are to account for the phenomena,